Universal Taylor series

A universal Taylor series is a formal power series ∑n=1∞anxn, such that for every continuous function h on [−1,1], if h(0)=0, then there exists an increasing sequence (λn) of positive integers such thatlimn->∞‖∑k=1λnakxk−h(x)‖=0In other words, the set of partial sums of ∑n=1∞anxn is dense (in sup-norm) in C[−1,1]0, the set of continuous functions on [−1,1] that is zero at origin.[1]

## Statements and proofs

Fekete proved that a universal Taylor series exists.[2]

Proof

Let f1,f2,... be the sequence in which each rational-coefficient polynomials with zero constant coefficient appears countably infinitely many times (use the diagonal enumeration). By Weierstrass approximation theorem, it is dense in C[−1,1]0. Thus it suffices to approximate the sequence. We construct the power series iteratively as a sequence of polynomials p1,p2,..., such that pn,pn+1 agrees on the first n coefficients, and ‖fn−pn‖∞≤1/n.

To start, let p1=f1. To construct pn+1, replace each x in fn+1−pn by a close enough approximation with lowest degree ≥n+1, using the lemma below. Now add this to pn.

Lemma -- The function f(x)=x can be approximated to arbitrary precision with a polynomial with arbitrarily lowest degree. That is, ∀ϵ>0,n∈{1,2,...}∃ polynomial p(x)=anxn+⋯+aNxN, such that ‖f−p‖∞≤ϵ.

Proof of lemma

The function g(x)=x−ctanh⁡(x/c) is the uniform limit of its Taylor expansion, which starts with degree 3. Also, ‖f−g‖∞<c. Thus to ϵ-approximate f(x)=x using a polynomial with lowest degree 3, we do so for g(x) with c<ϵ/2 by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of g(x), obtaining an approximation of lowest degree 9, 27, 81...

## References

1. ↑ Mouze, A.; Nestoridis, V. (2010). "Universality and ultradifferentiable functions: Fekete's theorem" (in en). Proceedings of the American Mathematical Society 138 (11): 3945–3955. doi:10.1090/S0002-9939-10-10380-3. ISSN 0002-9939. https://www.ams.org/proc/2010-138-11/S0002-9939-10-10380-3/.
2. ↑ Pál, Julius (1914). "Zwei kleine Bemerkungen". Tohoku Mathematical Journal. First Series 6: 42–43. https://www.jstage.jst.go.jp/article/tmj1911/6/0/6_0_42/_article/-char/ja/.

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